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The “center” of a data set is a way of describing location. The two most widely used measures of the “center” of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data, and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center. NOTEThe words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term for mean is “arithmetic mean” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.” MeanThe mean is calculated by adding up all of the values in the data and then dividing the sum by the total number of data values. The letter used to represent the sample mean is [latex]\displaystyle{\overline{x}}[/latex] (read [latex]x[/latex]-bar). The Greek letter [latex]\mu[/latex] (pronounced “mew”) represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random. Consider the sample:
[latex]\displaystyle{\overline{x} = \frac{1+1+1+2+2+3+4+4+4+4+4}{11} = 2.7}[/latex] To find the mean in Excel, use the average(array) function.
The output from the average function is the mean of the entered data. Visit the Microsoft page for more information about the average function. MedianThe median is the middle value in an orderedset of data. You can quickly find the location of the median by using the expression [latex]\displaystyle{\frac{n+1}{2}}[/latex] where [latex]n[/latex] is the total number of data values in the sample. If [latex]n[/latex] is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If [latex]n[/latex] is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then the median is located in position [latex]\displaystyle{\frac{n+1}{2}=\frac{97+1}{2}=49}[/latex] of the ordered list. If the total number of data values is 100, then [latex]\displaystyle{\frac{n+1}{2}=\frac{100+1}{2}=50.5}[/latex] and the median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The upper case letter [latex]M[/latex] is often used to represent the median. To find the median in Excel, use the median(array) function.
The output from the median function is the median of the entered data. Visit the Microsoft page for more information about the median function. AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):
Calculate the mean and the median. Solution: Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A40. For the mean:
For the median:
The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median.
Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A39. For the mean:
For the median:
Suppose that in a small town of 50 people, one person earns [latex]\$5,000,000[/latex] per year and the other 49 each earn [latex]\$30,000[/latex]. Which is the better measure of the “center”: the mean or the median? Solution: [latex]\displaystyle{\mu=\frac{5,000,000+(49 \times30,000)}{50}=129,400}[/latex] [latex]M=30,000[/latex] The median is a better measure of the “center” than the mean because 49 of the values are [latex]\$30,000[/latex] and one is [latex]\$5,000,000[/latex]. The [latex]\$5,000,000[/latex] is an outlier. The median of [latex]\$30,000[/latex] gives us a better sense of the middle of the data. In a sample of 60 households, one house is worth [latex]\$2,500,000[/latex]. Half of the rest are worth [latex]\$280,000[/latex], and all the others are worth [latex]\$315,000[/latex]. Which is the better measure of the “center”: the mean or the median? Click to see SolutionThe median is the better measure of the “center” than the mean because 59 of the values are either [latex]\$280,000[/latex] or [latex]\$315,000[/latex] and only one is [latex]\$2,500,000[/latex]. The [latex]\$2,500,000[/latex] is an outlier. Either [latex]\$280,000[/latex] or [latex]\$315,000[/latex] gives us a better sense of the middle of the data. ModeAnother measure of the center of the data is the mode. The mode is the most frequently occurring value in the set of data. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A set of data can also have no mode if all of the observations in the data are unique. Unlike the mean and the median, the mode can be calculated for both qualitative data and quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red. To find the mode in Excel:
Statistics exam scores for 20 students are as follows:
Find the mode. Solution: Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A20. Start by using the count function to count the number of modes in the data:
Because the output from the count(mode.mult(…)) function is 1, there is only 1 mode in the data. To find the single mode, we use the mode.sngl function:
By examining the data, we can see that 72 is the most frequently occurring value (5 times) and that 72 is the only value that occurs 5 times. The number of books checked out from the library from 25 students are as follows:
Find the mode. Click to see SolutionEnter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A25. Start by using the count function to count the number of modes in the data:
Because the output from the count(mode.mult(…)) function is 1, there is only 1 mode in the data. To find the single mode, we use the mode.sngl function:
The most frequent number of books is [latex]7[/latex], which occurs four times. AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):
Calculate the mode. Solution: Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A40. Start by using the count function to count the number of modes in the data:
Because the output from the count(mode.mult(…)) function is 12, there are 12 modes in the data. To find the multiple modes, we use the mode.mult function. Left click on a cell, hold and drag down to highlight 12 vertical cells. In the highlighted cells, enter the mode.mult function:
Because the output from the mode.mult function is a (vertical) array after entering the function press CTRL+SHIFT+ENTER (not ENTER by itself). Ten credit scores are
Calculate the mode. Click to see SolutionEnter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A10. Start by using the count function to count the number of modes in the data:
Because the output from the count(mode.mult(…)) function is 2, there are 2 modes in the data. To find the multiple modes, we use the mode.mult function. Left click on a cell, hold and drag down to highlight 2 vertical cells. In the highlighted cells, enter the mode.mult function:
Because the output from the mode.mult function is a (vertical) array after entering the function press CTRL+SHIFT+ENTER (not ENTER by itself). Watch this video: Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy by Khan Academy [3:54] (transcript available). Calculating the Mean of Grouped Frequency TablesWhen only grouped data is available, we do not know the individual data values (we only know intervals and interval frequencies). Therefore, we cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: [latex]\displaystyle{\text{mean}=\frac{\text{data sum}}{\text{number of data values}}}[/latex] We simply need to modify the definition to fit within the restrictions of a frequency table. Because we do not know the individual data values, we use the midpoint of each interval. The midpoint of an interval is [latex]\displaystyle{\mbox{midpoint}=\frac{\text{lower boundary } + \text{ upper boundary}}{2}}[/latex] We can now modify the mean definition to be [latex]\displaystyle{\text{Mean of Frequency Table} = \frac{\sum (f \times m)}{\sum f}}[/latex] where [latex]f[/latex] is the frequency of the interval and [latex]m[/latex] is the midpoint of the interval. A frequency table displaying professor Blount’s last statistic test is shown. Find the best estimate of the class mean.
Solution: Find the midpoints for all intervals
[latex]\begin{eqnarray*} \mbox{Mean} & = & \frac{53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1)}{19} \\ & = & \frac{1460.25}{19} \\& = & 78.86 \end{eqnarray*}[/latex] Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:
What is the best estimate for the mean number of hours spent playing video games? Click to the SolutionThe midpoints are [latex]1.75[/latex], [latex]5.5[/latex], [latex]9.5[/latex], [latex]13.5[/latex], [latex]17.5[/latex]. [latex]\begin{eqnarray*}\mbox{Mean} & = & \frac{(1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9)}{38} \\ & = & \frac{409.75}{38}\\ & = &10.78\end{eqnarray*}[/latex] When to Use Each Measure of Central TendencyThe measures of central tendency tell us about the center of the data, but often give different answers. So how do we know when to use each? Here are some general rules:
Concept ReviewThe mean, the median, and the mode are measures of the “center” of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode tells us the most frequently occurring datum (or data) in our data set. The mean, median, and mode are extremely helpful when we need to analyze our data, but if the data set consists of ranges which lack specific values, the mean may be impossible to calculate. However, the mean of grouped data can be approximated by multiplying the midpoint of each interval with the frequency, adding up these values and then dividing by the total number of values in the data set. Attribution“2.5 Measures of the Center of the Data“ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License. Which measure of central tendency is obtained by calculating the sum of values and diving this figure by the number of values there are in the data set?The mean is the sum of the value of each observation in a dataset divided by the number of observations. This is also known as the arithmetic average.
Which measure of central tendency is obtained by calculating the sum of values brainly?The arithmetic mean is the most common measure of central tendency. It is computed by summing all the scores (sigma or Σ) and dividing by the number of scores (N):
What measure is calculated by adding all the values and dividing the sum by the number of values?The mean of a data set is the sum of the values divided by the number of values. The median of a data set is the middle value when the values are written in numerical order.
Which measure of central tendency is computed by dividing the sum of the data by the number of pieces of data?The mean is often called the average. To find the mean you take a set of data and calculate the sum of the data, after that you divide the sum by the number of pieces in the set.
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